Versions 1 and 2 of the 9-faced toroidal polyhedra are locally regular (equivelar). The combinatorial structures of the two versions are identical. The third figure is different combinatorially; its polyhedron is globally regular.
A locally regular (or equivelar) torus-like polyhedron has the following properties: 1. Each face has the same number of sides. 2. The same number of edges meet at each vertex. 3. Two faces have at most two vertices in common, and if they share two vertices and , then the faces share the edge . This requirement prevents overarching faces.
A globally regular toroid is a locally regular toroid with the additional property that its topological automorphism group acts transitively on the flags. (A flag is a mutually incident triple of a vertex, an edge, and a face.)
 L. Szilassi, "On Three Classes of Regular Toroids," Symmetry: Culture and Science, 11(1–4), 2000 pp. 317–335.
 B. M. Stewart, Adventures among the Toroids, rev. 2nd ed., Okemos, MI: B. M. Stewart, 1980 p. 199.
Wolfram Demonstrations Project
Published: April 6 2016